Problem: Santa has an infinite number of candies for each of m flavours. You are given a rooted tree with n vertices. The root of the tree is the vertex 1. Each vertex contains exactly one candy. The i-th vertex has a candy of flavour f_i.

Sometimes Santa fears that candies of flavour k have melted. He chooses any vertex x randomly and sends the subtree of x to the Bakers for a replacement. In a replacement, all the candies with flavour k are replaced with a new candy of the same flavour. The candies which are not of flavour k are left unchanged. After the replacement, the tree is restored.

The actual cost of replacing one candy of flavour k is c_k (given for each k). The Baker keeps the price fixed in order to make calculation simple. Every time when a subtree comes for a replacement, the Baker charges C, no matter which subtree it is and which flavour it is.

Suppose that for a given flavour k the probability that Santa chooses a vertex for replacement is same for all the vertices. You need to find out the expected value of error in calculating the cost of replacement of flavour k. The error in calculating the cost is defined as follows.

$$$ Error\ E(k) =\ (Actual Cost\ –\ Price\ charged\ by\ the\ Bakers) ^ 2.$$$

Note that the actual cost is the cost of replacement of one candy of the flavour k multiplied by the number of candies in the subtree.

Also, sometimes Santa may wish to replace a candy at vertex x with a candy of some flavour from his pocket.

You need to handle two types of operations: 

  * Change the flavour of the candy at vertex x to w. 
  * Calculate the expected value of error in calculating the cost of replacement for a given flavour k. 

Input

The first line of the input contains four integers n (2 ⩽ n ⩽ 5 ⋅ 10^4), m, q, C (1 ⩽ m, q ⩽ 5 ⋅ 10^4, 0 ⩽ C ⩽ 10^6) — the number of nodes, total number of different flavours of candies, the number of queries and the price charged by the Bakers for replacement, respectively.

The second line contains n integers f_1, f_2, ..., f_n (1 ⩽ f_i ⩽ m), where f_i is the initial flavour of the candy in the i-th node.

The third line contains n - 1 integers p_2, p_3, ..., p_n (1 ⩽ p_i ⩽ n), where p_i is the parent of the i-th node.

The next line contains m integers c_1, c_2, ... c_m (1 ⩽ c_i ⩽ 10^2), where c_i is the cost of replacing one candy of flavour i.

The next q lines describe the queries. Each line starts with an integer t (1 ⩽ t ⩽ 2) — the type of the query.

If t = 1, then the line describes a query of the first type. Two integers x and w follow (1 ⩽ x ⩽ n, 1 ⩽ w ⩽ m), it means that Santa replaces the candy at vertex x with flavour w.

Otherwise, if t = 2, the line describes a query of the second type and an integer k (1 ⩽ k ⩽ m) follows, it means that you should print the expected value of the error in calculating the cost of replacement for a given flavour k.

The vertices are indexed from 1 to n. Vertex 1 is the root.

Output

Output the answer to each query of the second type in a separate line.

Your answer is considered correct if its absolute or relative error does not exceed 10^{-6}.

Formally, let your answer be a, and the jury's answer be b. The checker program considers your answer correct if and only if (|a-b|)/(max(1,b))⩽ 10^{-6}.

Example

Input

3 5 5 7
3 1 4
1 1
73 1 48 85 89
2 1
2 3
1 2 3
2 1
2 3


Output

2920.333333333333
593.000000000000
49.000000000000
3217.000000000000

Note

For 1-st query, the error in calculating the cost of replacement for flavour 1 if vertex 1, 2 or 3 is chosen are 66^2, 66^2 and (-7)^2 respectively. Since the probability of choosing any vertex is same, therefore the expected value of error is (66^2+66^2+(-7)^2)/(3).

Similarly, for 2-nd query the expected value of error is (41^2+(-7)^2+(-7)^2)/(3).

After 3-rd query, the flavour at vertex 2 changes from 1 to 3.

For 4-th query, the expected value of error is ((-7)^2+(-7)^2+(-7)^2)/(3).

Similarly, for 5-th query, the expected value of error is (89^2+41^2+(-7)^2)/(3).

Solution (in JAVA):

Solution:
import static java.lang.Integer.parseInt;
import static java.lang.Long.parseLong;
import static java.lang.Math.max;
import static java.lang.System.arraycopy;
import static java.lang.System.exit;
import static java.util.Arrays.copyOf;

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.Locale;
import java.util.NoSuchElementException;
import java.util.StringTokenizer;

public class H {

	static class IntList {

		int data[] = new int[3];
		int size = 0;

		boolean isEmpty() {
			return size == 0;
		}

		int size() {
			return size;
		}

		int get(int index) {
			if (index < 0 || index >= size) {
				throw new IndexOutOfBoundsException();
			}
			return data[index];
		}

		void clear() {
			size = 0;
		}

		void set(int index, int value) {
			if (index < 0 || index >= size) {
				throw new IndexOutOfBoundsException();
			}
			data[index] = value;
		}

		void expand() {
			if (size >= data.length) {
				data = copyOf(data, (data.length << 1) + 1);
			}
		}

		void insert(int index, int value) {
			if (index < 0 || index > size) {
				throw new IndexOutOfBoundsException();
			}
			expand();
			arraycopy(data, index, data, index + 1, size++ - index);
			data[index] = value;
		}

		int delete(int index) {
			if (index < 0 || index >= size) {
				throw new IndexOutOfBoundsException();
			}
			int value = data[index];
			arraycopy(data, index + 1, data, index, --size - index);
			return value;
		}

		void push(int value) {
			expand();
			data[size++] = value;
		}

		int pop() {
			if (size == 0) {
				throw new NoSuchElementException();
			}
			return data[--size];
		}

		void unshift(int value) {
			expand();
			arraycopy(data, 0, data, 1, size++);
			data[0] = value;
		}

		int shift() {
			if (size == 0) {
				throw new NoSuchElementException();
			}
			int value = data[0];
			arraycopy(data, 1, data, 0, --size);
			return value;
		}
	}

	static class LongLongMap {

		static final long EMPTY = Long.MIN_VALUE;

		long data[] = new long[4];
		{
			data[0] = EMPTY;
			data[2] = EMPTY;
		}
		int size = 0;

		boolean isEmpty() {
			return size == 0;
		}

		int size() {
			return size;
		}

		void clear() {
			for (int i = 0; i < data.length; i += 2) {
				data[i] = EMPTY;
			}
			size = 0;
		}

		static int hash(long key) {
			long hash = key * 1000000007;
			hash ^= hash >> 16;
			hash ^= hash >> 24;
			return (int) hash;
		}

		boolean have(long key) {
			if (key == EMPTY) {
				throw new IllegalArgumentException();
			}
			for (int mask = data.length - 2, i = hash(key) & mask;; i = (i - 1) & mask) {
				long cur = data[i];
				if (cur == key) {
					return true;
				} else if (cur == EMPTY) {
					return false;
				}
			}
		}

		long get(long key, long default_) {
			if (key == EMPTY) {
				throw new IllegalArgumentException();
			}
			for (int mask = data.length - 2, i = hash(key) & mask;; i = (i - 1) & mask) {
				long cur = data[i];
				if (cur == key) {
					return data[i + 1];
				} else if (cur == EMPTY) {
					return default_;
				}
			}
		}

		static long[] resize(long data[], int newCapacity) {
			int oldCapacity = data.length;
			long ndata[] = new long[newCapacity];
			for (int i = 0; i < newCapacity; i += 2) {
				ndata[i] = EMPTY;
			}
			for (int mask = newCapacity - 2, i = 0; i < oldCapacity; i += 2) {
				long cur = data[i];
				if (cur == EMPTY) {
					continue;
				}
				for (int j = hash(cur) & mask;; j = (j - 1) & mask) {
					if (ndata[j] == EMPTY) {
						ndata[j] = cur;
						ndata[j + 1] = data[i + 1];
						break;
					}
				}
			}
			return ndata;
		}

		void set(long key, long value) {
			if (key == EMPTY) {
				throw new IllegalArgumentException();
			}
			long data[] = this.data;
			for (int capacity = data.length, mask = capacity - 2, i = hash(key) & mask;; i = (i - 1) & mask) {
				long cur = data[i];
				if (cur == key) {
					data[i + 1] = value;
					return;
				} else if (cur == EMPTY) {
					data[i] = key;
					data[i + 1] = value;
					if (++size > capacity >> 2) {
						this.data = resize(data, capacity << 1);
					}
					return;
				}
			}
		}

		void delete(long key) {
			if (key == EMPTY) {
				throw new IllegalArgumentException();
			}
			long data[] = this.data;
			for (int capacity = data.length, mask = capacity - 2, i = hash(key) & mask;; i = (i - 1) & mask) {
				long cur = data[i];
				if (cur == key) {
					for (int j = i;;) {
						cur = data[j = (j - 1) & mask];
						if (cur == EMPTY) {
							data[i] = EMPTY;
							--size;
							return;
						}
						int k = hash(cur) & mask;
						if ((k >= i) ^ (k < j) ^ (i < j)) {
							data[i] = cur;
							data[i + 1] = data[j + 1];
							i = j;
						}
					}
				} else if (cur == EMPTY) {
					return;
				}
			}
		}

		void expand(int newCapacity) {
			long data[] = this.data;
			int oldCapacity = data.length;
			if (oldCapacity >> 2 < newCapacity) {
				this.data = resize(data, Integer.highestOneBit(newCapacity - 1) << 3);
			}
		}

		void shrink() {
			long data[] = this.data;
			int oldCapacity = data.length;
			if (oldCapacity > 4 && oldCapacity >> 3 >= size) {
				this.data = resize(data, max(Integer.highestOneBit(size - 1) << 3, 4));
			}
		}

		static class Iterator {

			final long data[];
			int position = -2;

			Iterator(long data[]) {
				this.data = data;
				advance();
			}

			boolean hasNext() {
				return position < data.length;
			}

			long getKey() {
				return data[position];
			}

			long getValue() {
				return data[position + 1];
			}

			void advance() {
				do {
					position += 2;
				} while (position < data.length && data[position] == EMPTY);
			}
		}

		Iterator iterator() {
			return new Iterator(data);
		}
	}

	static class LongLongCounter extends LongLongMap {

		long get(long key) {
			return get(key, 0);
		}

		void set(long key, long value) {
			if (value == 0) {
				delete(key);
			} else {
				super.set(key, value);
			}
		}

		void add(long key, long value) {
			if (key == EMPTY) {
				throw new IllegalArgumentException();
			}
			long data[] = this.data;
			for (int capacity = data.length, mask = capacity - 2, i = hash(key) & mask;; i = (i - 1) & mask) {
				long cur = data[i];
				if (cur == key) {
					data[i + 1] += value;
					return;
				} else if (cur == EMPTY) {
					data[i] = key;
					data[i + 1] = value;
					if (++size > capacity >> 2) {
						this.data = resize(data, capacity << 1);
					}
					return;
				}
			}
		}
	}

	static long getSum(LongLongCounter fenwick, int i) {
		long res = fenwick.get(i);
		while ((i = (i & (i + 1)) - 1) >= 0) {
			res += fenwick.get(i);
		}
		return res;
	}

	static void updateSum(LongLongCounter fenwick, int i, long v) {
		fenwick.add(i, v);
		while ((i |= i + 1) < 60000) {
			fenwick.add(i, v);
		}
	}

	static void dfs1(int cur, int cDepth) {
		depth[cur] = cDepth;
		tEnter[cur] = tCur++;
		for (int i = 0; i < children[cur].size(); i++) {
			dfs1(children[cur].get(i), cDepth + 1);
		}
		tExit[cur] = tCur;
	}

	static void dfs2(int cur, int cHpRoot, int cHpPos) {
		hpRoot[cur] = cHpRoot;
		hpPos[cur] = cHpPos;
		int heavy = -1, heavySize = -1;
		for (int i = 0; i < children[cur].size(); i++) {
			int next = children[cur].get(i);
			if (heavySize < tExit[next] - tEnter[next]) {
				heavy = next;
				heavySize = tExit[next] - tEnter[next];
			}
		}
		hpChild[cur] = heavy;
		for (int i = 0; i < children[cur].size(); i++) {
			int next = children[cur].get(i);
			if (next == heavy) {
				dfs2(next, cHpRoot, cHpPos + 1);
			} else {
				dfs2(next, next, 0);
			}
		}
		if (cur == cHpRoot) {
			int cnt = 1;
			for (int i = heavy; i >= 0; i = hpChild[i]) {
				++cnt;
			}
			hpList[cur] = new int[cnt];
			cnt = 0;
			int i = cur;
			do {
				hpList[cur][cnt] = i;
				i = hpChild[i];
				++cnt;
			} while (i >= 0);
		}
	}

	static void put(int i, int f, int w) {
		int add = w * depth[i];
		count1[f] += add;
		updateSum(fw1[f], tEnter[i], w);
		count2[f] += add * (getSum(fw1[f], tExit[i] - 1) - getSum(fw1[f], tEnter[i]));
		while (true) {
			int curRoot = hpRoot[i], curPos = hpPos[i], curList[] = hpList[curRoot], curLimit = curList.length;
			do {
				fw2[f].add(curList[curPos], add);
				curPos |= curPos + 1;
			} while (curPos < curLimit);
			curPos = hpPos[i] - 1;
			while (curPos >= 0) {
				count2[f] += w * fw2[f].get(curList[curPos]);
				curPos = (curPos & (curPos + 1)) - 1;
			}
			i = parent[curRoot];
			if (i < 0) {
				break;
			}
			add = w * depth[i];
			count2[f] += add * getSum(fw1[f], tExit[i] - 1);
			if (tEnter[i] > 0) {
				count2[f] -= add * getSum(fw1[f], tEnter[i] - 1);
			}
			count2[f] -= add * (getSum(fw1[f], tExit[curRoot] - 1) - getSum(fw1[f], tEnter[curRoot] - 1));
		}
	}

	static int parent[];
	static IntList children[];
	static int depth[], tCur, tEnter[], tExit[];
	static int hpRoot[], hpPos[], hpChild[], hpList[][];
	static LongLongCounter fw1[], fw2[];
	static long count1[], count2[];

	static void solve() throws Exception {
		int n = scanInt();
		int m = scanInt();
		int q = scanInt();
		int cBig = scanInt();
		int f[] = new int[n];
		for (int i = 0; i < n; i++) {
			f[i] = scanInt() - 1;
		}
		parent = new int[n];
		parent[0] = -1;
		children = new IntList[n];
		for (int i = 0; i < n; i++) {
			children[i] = new IntList();
		}
		for (int i = 1; i < n; i++) {
			parent[i] = scanInt() - 1;
			children[parent[i]].push(i);
		}
		depth = new int[n];
		tCur = 0;
		tEnter = new int[n];
		tExit = new int[n];
		dfs1(0, 1);
		hpRoot = new int[n];
		hpPos = new int[n];
		hpChild = new int[n];
		hpList = new int[n][];
		dfs2(0, 0, 0);
		fw1 = new LongLongCounter[m];
		fw2 = new LongLongCounter[m];
		for (int i = 0; i < m; i++) {
			fw1[i] = new LongLongCounter();
			fw2[i] = new LongLongCounter();
		}
		count1 = new long[m];
		count2 = new long[m];
		for (int i = 0; i < n; i++) {
			put(i, f[i], 1);
		}
		int c[] = new int[m];
		for (int i = 0; i < m; i++) {
			c[i] = scanInt();
		}
		for (int qq = 0; qq < q; qq++) {
			switch (scanInt()) {
			case 1:
				int x = scanInt() - 1;
				int w = scanInt() - 1;
				put(x, f[x], -1);
				put(x, f[x] = w, 1);
				break;
			case 2:
				int k = scanInt() - 1;
				long sum = count1[k];
				long sum2 = sum + 2 * count2[k];
				double ans = (double) (c[k] * c[k] * sum2 - 2 * c[k] * sum * cBig) / n + (long) cBig * cBig;
				out.printf(Locale.US, "%.9f\n", ans);
				break;
			default:
				throw new AssertionError();
			}
		}
	}

	static int scanInt() throws IOException {
		return parseInt(scanString());
	}

	static long scanLong() throws IOException {
		return parseLong(scanString());
	}

	static String scanString() throws IOException {
		while (tok == null || !tok.hasMoreTokens()) {
			tok = new StringTokenizer(in.readLine());
		}
		return tok.nextToken();
	}

	static BufferedReader in;
	static PrintWriter out;
	static StringTokenizer tok;

	public static void main(String[] args) {
		try {
			in = new BufferedReader(new InputStreamReader(System.in));
			out = new PrintWriter(System.out);
			solve();
			in.close();
			out.close();
		} catch (Throwable e) {
			e.printStackTrace();
			exit(1);
		}
	}
}